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A generalization of Anderson's theorem on unimodal functions


Author: Somesh Das Gupta
Journal: Proc. Amer. Math. Soc. 60 (1976), 85-91
MSC: Primary 26A87; Secondary 52A40
DOI: https://doi.org/10.1090/S0002-9939-1976-0425050-7
MathSciNet review: 0425050
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Abstract: Anderson (1955) gave a definition of a unimodal function on $ {R^n}$ and obtained an inequality for integrals of a symmetric unimodal function over translates of a symmetric convex set. Anderson's assumptions, especially the role of unimodality, are critically examined and generalizations of his inequality are obtained in different directions. It is shown that a marginal function of a unimodal function (even if it is symmetric) need not be unimodal.


References [Enhancements On Off] (What's this?)

  • [T] W. Anderson (1955), The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities, Proc. Amer. Math. Soc. 6, 170-176. MR 16, 1005. MR 0069229 (16:1005a)
  • [G] S. Mudholkar (1966), The integral of an invariant unimodal function over an invariant convex set--An inequality and applications, Proc. Amer. Math. Soc. 17, 1327-1333. MR 34 #7741. MR 0207928 (34:7741)
  • [S] Sherman (1955), A theorem on convex sets with applications, Ann. Math. Statist. 26, 763-767. MR 17, 655. MR 0074845 (17:655l)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0425050-7
Keywords: Unimodal function, convex set, invariance, marginal function, inequalities
Article copyright: © Copyright 1976 American Mathematical Society

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